| 1. |
- Hedenmalm, Håkan, 1961-, et al.
(författare)
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Heisenberg uniqueness pairs and the Klein-Gordon equation
- 2011
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Ingår i: Annals of Mathematics. - : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 173:3, s. 1507-1527
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Tidskriftsartikel (refereegranskat)abstract
- A Heisenberg uniqueness pair (HUP) is a pair (Γ,Λ), where Γ is a curve in the plane and Λ is a set in the plane, with the following property: any finite Borel measure μ in the plane supported on Γ, which is absolutely continuous with respect to arc length, and whose Fourier transform μˆ vanishes on Λ, must automatically be the zero measure. We prove that when Γ is the hyperbola x1x2=1 %, and Λ is the lattice-cross Λ=(αZ×{0})∪({0}×βZ), where α,β are positive reals, then (Γ,Λ) is an HUP if and only if αβ≤1; in this situation, the Fourier transform μˆ of the measure solves the one-dimensional Klein-Gordon equation. Phrased differently, we show that eπiαnt,eπiβn/t,n∈Z, span a weak-star dense subspace in L∞(R) if and only if αβ≤1. In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoff Ergodic Theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.
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| 2. |
- Aleman, Alexandru, et al.
(författare)
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Backward Shift and Nearly Invariant Subspaces of Fock-type Spaces
- 2022
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Ingår i: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2022:10, s. 7390-7419
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Tidskriftsartikel (refereegranskat)abstract
- We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces ℱWp, whose weight is not necessarily radial. We show that in the spaces ℱWp, which contain the polynomials as a dense subspace (in particular, in the radial case), all nontrivial backward shift invariant subspaces are of the form ℘n, that is, finite-dimensional subspaces consisting of polynomials of degree at most n. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type), we establish an analogue of de Branges' ordering theorem. We then construct examples that show that the result fails for general Fock-type spaces of larger growth.
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| 3. |
- Bakan, Andrew, et al.
(författare)
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Exponential Integral Representations of Theta Functions
- 2020
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Ingår i: Computational methods in Function Theory. - : Springer. - 1617-9447 .- 2195-3724. ; 20:3-4, s. 591-621
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Tidskriftsartikel (refereegranskat)abstract
- Let Θ3(z) : = ∑ n∈Zexp (i πn2z) be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane H:={z∈C|Imz>0}, and takes positive values along i R> 0, the positive imaginary axis, where R> 0: = (0 , + ∞). We define its logarithm log Θ3(z) which is uniquely determined by the requirements that it should be holomorphic in H and real-valued on i R> 0. We derive an integral representation of log Θ3(z) when z belongs to the hyperbolic quadrilateral F□||:={z∈C|Imz>0,-1≤Rez≤1,|2z-1|>1,|2z+1|>1}.Since every point of H is equivalent to at least one point in F□|| under the theta subgroup of the modular group on the upper half-plane, this representation carries over in modified form to all of H via the identity recorded by Berndt. The logarithms of the related Jacobi theta functions Θ4 and Θ2 may be conveniently expressed in terms of log Θ3 via functional equations, and hence get controlled as well. Our approach is based on a study of the logarithm of the Gauss hypergeometric function for a specific choice of the parameters. This has connections with the study of the universally starlike mappings introduced by Ruscheweyh, Salinas, and Sugawa.
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| 4. |
- Bakan, Andrew, et al.
(författare)
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Fourier uniqueness in even dimensions
- 2021
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Ingår i: Proceedings of the National Academy of Sciences of the United States of America. - : Proceedings of the National Academy of Sciences. - 0027-8424 .- 1091-6490. ; 118:15
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Tidskriftsartikel (refereegranskat)abstract
- In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions (d = 1, 8, 24), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions d = 8 and d = 24. In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the KleinGordon equation. Since the existing method for the Klein-Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein-Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal.
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| 6. |
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| 7. |
- Hedenmalm, Håkan, 1961-
(författare)
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A Beurling-Rudin theorem for H^\infty
- 1987
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Ingår i: Illinois Journal of Mathematics. - 0019-2082 .- 1945-6581. ; 31, s. 629-644
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Tidskriftsartikel (refereegranskat)
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| 8. |
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| 10. |
- Hedenmalm, Håkan, 1961-, et al.
(författare)
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A critical topology for L^p Carleman classes with 0
- 2018
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 371:3-4, s. 1803-1844
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we explore a sharp phase transition phenomenon which occurs for (Formula presented.)-Carleman classes with exponents (Formula presented.). These classes are defined as for the standard Carleman classes, only the (Formula presented.)-bounds are replaced by corresponding (Formula presented.)-bounds. We study the quasinorms (Formula presented.)for some weight sequence (Formula presented.) of positive real numbers, and consider as the corresponding (Formula presented.)-Carleman space the completion of a given collection of smooth test functions. To mirror the classical definition, we add the feature of dilatation invariance as well, and consider a larger soft-topology space, the (Formula presented.)-Carleman class. A particular degenerate instance is when (Formula presented.) for (Formula presented.) and (Formula presented.) for (Formula presented.). This would give the (Formula presented.)-Sobolev spaces, which were analyzed by Peetre, following an initial insight by Douady. Peetre found that these (Formula presented.)-Sobolev spaces are highly degenerate for (Formula presented.). Indeed, the canonical map (Formula presented.) fails to be injective, and there is even an isomorphism (Formula presented.)corresponding to the canonical map (Formula presented.) acting on the test functions. This means that e.g. the function and its derivative lose contact with each other (they “disconnect”). Here, we analyze this degeneracy for the more general (Formula presented.)-Carleman classes defined by a weight sequence (Formula presented.). If (Formula presented.) has some regularity properties, and if the given collection of test functions is what we call (Formula presented.)-tame, then we find that there is a sharp boundary, defined in terms of the weight (Formula presented.): on the one side, we get Douady–Peetre’s phenomenon of “disconnexion”, while on the other, the completion of the test functions consists of (Formula presented.)-smooth functions and the canonical map (Formula presented.) is correspondingly well-behaved in the completion. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the (Formula presented.) setting, with (Formula presented.).
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